Martians

A sphere-shaped spaceship with a diameter of 6 m landed in the meadow. In order not to attract attention, the Martians covered it with a roof in the shape of a regular cone. How high will this roof be so that the consumption of roofing is minimal?

Correct answer:

S =  115.6965 m²

Step-by-step explanation:

$$\begin{gathered} D=6\text{ m } \\ R=D\mathrm{/}2=6\mathrm{/}2=3\text{ m } \\ \sin \alpha =r\mathrm{/}s \\ R:(h-R)=r:s=\sin \alpha \\ r=s\cdot {\displaystyle \frac{R}{h-R}} \\ S=\pi \cdot r\cdot s=\pi \cdot R\mathrm{/}(h-R)\cdot s^2 \\ S=\pi \cdot R\mathrm{/}(h-R)\cdot ((R\mathrm{/}(h-R){)}^2+h^2) \\ S=(3(9\mathrm{/}(-3+h{)}^2+h^2)\pi )\mathrm{/}(-3+h) \\ h=6.37252\doteq 6.3725\text{ m } \\ s=\sqrt{(R\mathrm{/}(h-R){)}^2+h^2}=\sqrt{(3\mathrm{/}(6.3725-3){)}^2+6.3725^2}\doteq 6.4343\text{ m } \\ r=s\cdot {\displaystyle \frac{R}{h-R}}=6.4343\cdot {\displaystyle \frac{3}{6.3725-3}}\doteq 5.7236\text{ m } \\ S=\pi \cdot r\cdot s=3.1416\cdot 5.7236\cdot 6.4343=115.6965{\text{m}}^2 \end{gathered}$$

Try another example

Related math problems and questions:

• Cone roof
  How many m² of roofing is needed to cover a cone-shaped roof with a diameter of 10 m and a height of 4 m? Add an extra 4% to the overlays.
• Pyramid roof
  2/4 of the area of ​​the roof-shaped regular tetrahedral pyramid with base edge 10 m and height of 4 m is already covered with roofing. How many square meters still need to be covered?
• Church roof
  The roof of the church tower has the shape of a regular tetrahedral pyramid with base edge length 5.4 meters and a height 5 m. It was found that needs to be corrected 27% covering of the roof area. What amount of material will be required?
• Maximum of volume
  The shell of the cone is formed by winding a circular section with a radius of 1. For what central angle of a given circular section will the volume of the resulting cone be maximum?
• Storm and roof
  The roof on the building is a cone with a height of 3 meters and a radius equal to half the height of the roof. How many m² of roof need to be repaired if 20% were damaged in a storm?
• The roof of the church
  The cone roof of the church has a diameter of 3 m and a height of 4 m. What is the size of the side edge of the church roof (s=?) and many sheets sheet will be needed to cover the church roof?
• Sphere and cone
  Within the sphere of radius G = 33 cm inscribe the cone with the largest volume. What is that volume, and what are the dimensions of the cone?
• The roof
  The roof has the shape of a spherical canopy with a base diameter of 8 m and a height of 2 m, calculate the area of the foil with which the roof is covered, when we calculate 13% for waste and residues.
• Roof 7
  The roof has the shape of a regular quadrangular pyramid with a base edge of 12 m and a height of 4 m. How many percent is folds and waste if in construction was consumed 181.4m² of plate?
• Roof cover
  Above the pavilion with a square ground plan with a side length of a = 12 m is a pyramid-shaped roof with a height v = 4.5 m. Calculate how much m² of sheet metal is needed to cover this roof if 5.5% of the sheet we must add for joints and waste.
• Secret treasure
  Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Find the container's radius r (and height h) so that they can hide the largest possible treasure.
• Iglu - cone tent
  The cone-shaped tent is 3 m high, the diameter of its base is 3.2 m. a) The tent is made of two layers of material. How many m² of fabric is needed for production (including flooring), if 20% needs to be added to the minimum amount due to cutting waste? b
• The roof
  The roof of the tower has the shape of a regular quadrangular pyramid, the base edge of which is 11 m long and the side wall of the animal with the base an angle of 57°. Calculate how much roofing we need to cover the entire roof, if we count on 15% waste
• The bus stop
  The bus stop waiting room has the shape of a regular quadrilateral pyramid 4 m high with a 5 m base edge. Calculate how much m² roofing is required to cover the sheathing three walls, taking 40% of the additional coverage.
• Goat
  Meadow is a circle with a radius r = 19 m. How long must a rope tie a goat to the pin on the perimeter of the meadow to allow the goat to eat half of the meadow?
• Carpet
  The room is 10 x 5 meters. You have the role of carpet width of 1 meter. Make rectangular cut of roll that piece of carpet will be longest possible and it fit into the room. How long is a piece of carpet? Note .: carpet will not be parallel with the diago
• Castle tower
  The castle tower has a cone-shaped roof with a diameter of 10 meters and a height of 8 meters. Calculate how much m² of coverage is needed to cover it if we add one-third to the overlap.